newsletter109 v 2corkmaths.ie/docs/general/newsletter109.pdfpage 2 imta newsletter 109, 2009 welcome...

IMTA Newsletter 109, 2009

CONTENTS Editorial

Problem Solving in Second Level Mathematics —The Finnish Experience

—Erkki Pehkonen

A Practical Problem?

Groups and the Mosaics of the Alhambra—Neil Hallinan

Irish Junior Mathematics Competition Entry Form—Michael Moynihan

School Geometry—Anthony G. O’Farrell

Maths Week

The Importance of being Beautiful in Mathematics—Fiacre Ó Cairbre

Multiplication in Algebra Revisited—John Courlander

A Summary Of Efficient Numerical Tests Of Robin’s Reformulation Of The Riemann

Hypothesis – A Young Scientist Project

—Gary Carr, Graham McGrath, Daragh Moriarty

Simple Cables—A Young Scientist Project—Helen Iliff

The First International GeoGebra Conference—David Hobson

Just For Fun—John Courlander

Ireland and the IMO (1988-2009) —Gordon Lessells

Solutions to Leaving Certificate Mathematics Higher Level 2009—Maurice O’Driscoll

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IMTA COUNCIL 2009 Chairperson: Dominic Guinan Vice-Chair: John McArdle Correspondence: Donal Coughlan Recording Secretary: Sinead Breen Treasurer: Michael Moynihan Newsletter Editor: Neil Hallinan Second Level: Branch members nominated at AGM Third Level: As nominated at AGM—maximum 4 First Level: As nominated at AGM—1 member Syllabus Committee Rep.: Donal Coughlan Co-opted: Maximum 2 members A full list of Council is available on the IMTA website www.imta.ie This site also contains mate-rial relevant to the annual Team Maths quiz.

Cumann Oidí na h-Éireann Irish Mathematics Teachers’ Association

Founded in 1964 to promote and assist the teaching of mathematics at all levels. Membership is open to all those interested in mathematics and mathematics education. The IMTA is represented on all NCCA mathematics course committees. Individual membership may be obtained through a Branch organisation.

Acknowledgement The IMTA would like to thank The Celtic Press for contributing to the cost of producing this Newsletter.


Welcome to the Autumn 2009 edition of the IMTA Newsletter. Quite a lot has changed since this time last year. We are now in the advent of Project Maths as a practical reality in all our schools. In anticipation of this the IMTA was delighted to hear the thoughts and experiences of Profes-sor Erkki Pehkonen from the University of Helsinki, Finland, who delivered the Fr. Ingram Memorial Lecture at the AGM last November. As has been the custom over the last number of years the script of the Fr. Ingram Lecture is reproduced here. Professor Pehkonen’s conclusions are based on extensive research which is well referenced in this article. Other articles in this publication also show the results of erudite research. From NUI, Maynooth, Professor Anthony G. O’Farrell details the philosophical basis of the new Geometry syllabus which forms part of Strand 2 in Project Maths. This has been accepted by the syllabus committees and Board of Studies for Maths. Also from NUI, Maynooth, Dr. Fiacre Ó Cairbre presents a nuanced account of one of the most important aspects of mathematics—its connection with the concept of ‘beauty’. It is, perhaps, a necessary antidote to the more prevalent utilitarian view of mathematics that many imagine may underpin the new approach to mathematics teaching. Beauty and the appreciation of beauty in mathematics are driving forces much more potent than is often realised. Through the article on wallpaper groups, yours truly, Neil Hallinan, attempts to indicate how the beauty of mathe-matical constructs which was utilised by the Moorish residents in the Alhambra over seven hundred years ago may be made even more accessible through mathematical analysis. The flavour of Project Maths is continued with a relevant update on GeoGebra by David Hobson who reports on the most recent GeoGebra Conference in Hagenburg, Austria—the first such International Conference. GeoGe-bra is a dynamic mathematics software programme (freely distributed) which will, no doubt, gain in signifi-cance in the near future. Familiarity with such packages is a stated objective of the new Geometry syllabus. Gordon Lessells from the University of Limerick anno-tates the history of the International Mathematical Olympiad (IMO) in Ireland and the laudable achieve-ments of its Irish participants. In this edition we are also celebrating the work of some younger contributors who presented their work at the Young Scientist Exhibition 2009. Special thanks to Maurice O’Driscoll for the Solutions to

Editorial the Leaving Certificate 2009 Higher Level Mathemat-ics questions. We constantly appreciate the contribu-tion of this valuable resource. Congratulations to the winners of Team Maths, 2009, from Middleton, Co. Cork and to the runners-up from Marists, Athlone. A hearty congratulations, too, to all recently retired members but especially to our past Chairman, Liam Sayers, whose encouragement and wisdom were an invaluable input into the production of this Newslet-ter. Health and long years of enjoyment be yours! With thanks to all contributors. Enjoy the read! All contributions are welcome. Send by e-mail to : [email protected] or St. Mary’s, Holy Faith, Glasnevin, Dublin 11. Neil Hallinan Addendum: This Newsletter is not funded by the Teacher Education Section (TES) of the DES (Department of Education and Science.

Branches : Contacts Cork (Sec.): Brendan O’Sullivan, [email protected] Donegal (Sec.): Joe English, [email protected] Dublin (Sec.): Barbara Grace, [email protected] Galway (Tr.): Mary McMullin, [email protected] Kerry (Tr.): John O’Flaherty, [email protected] Limerick (Chair): Gary Ryan, [email protected] Mayo (Sec.): Lauranne Kelly, [email protected] Midlands (Sec.): Teresa Cushen, [email protected] Tipperary (Chair): Donal Coughlan, [email protected] Wexford : (Rep.): Sean MacCormaic, [email protected] Michael Brennan (Waterford) is still available for contact at [email protected]

The views expressed in this Newsletter are those of the individual authors and do not necessarily reflect the position of the IMTA. While every care has been taken to ensure that the information in this publication is up-to-date and correct no responsibility will be taken by the IMTA for any errors that might occur.

We note with regret the passing of Victor Bond, one of the “greats” of the IMTA, who contributed much in particular to its foundation and early years. We offer our sincere condolences to his family.


Problem Solving in Second Level Mathematics —The Finnish Experience

Fr. Ingram Memorial Lecture - 28th November, 2008 Erkki Pehkonen University of Helsinki

Abstract

Since problem solving is also internationally recognized to be a fuzzy concept, the presentation begins with a working characterization for a problem and some examples. The main content of the paper is to describe problem solving in Finnish school mathematics. This begins with consid-ering Finnish mathematics curricula with the focus on the role of problem solving. Furthermore, different manifestations problem solving in mathematics textbooks are discussed. And it will be described how Finnish teachers implement problem solving in mathematics lessons. Additionally the way teachers use problem solving in assessment is discussed briefly. At the end of the paper a solution for teaching reform within the curriculum is dealt with; the reform is based on the use of open problems. Finally some evaluation on the success of teaching reform in problem solving is given.

The purpose of school education in each country is, more or less, to develop independent, self-confident, critically thinking, motivated and multitalented individuals who will manage in societal settings that they will encounter later on in their life. The key question is what kind of school instruction is optimal for this goal.

Liam Sayers, IMTA Chair; Dr. Erkki Pehkonen; Dr. Maurice O’Reilly, Seminar Chair; Maurice O’Driscoll


PROBLEM SOLVING IN SCHOOL MATHEMATICS Problem solving has generally been accepted as a means for advancing thinking skills (e.g. Schoenfeld 1985). For example, in the NCTM Standards it is stated: Solving problems is not only a goal of learning mathematics but also a major means of doing so. … In everyday life and in the workplace, being a good problem solver can lead to great advantages. … Problem solving is an integral part of all mathematics learning. (NCTM, 2000, 52)

Yet, the basic concepts 'problem' and 'problem solving' seem still to be rather ambiguous in mathematics education. Sometimes a simple arithmetic task is understood as a 'problem' although it can be solved in a routine way, whereas at other times only more complex situations are considered as problems. The conception of mathematics underlying the definition is probably essential, but this conception is unfortunately seldom explicitly expressed. The theories of learning behind the practices of teaching can also be very different. Therefore, there might be big differences in the conceptions of problems and problem solving between different countries, and perhaps, also within a country. For example, some distinct differences in emphasis have been singled out between the American and the British conceptions (Lingard & al. 1985). Thus, for a paper to describe the implementation of problem solving in one particular country, it is important to explain the conception of problem solving within that country or at least its author’s conception.

Characterization of a problem Here we will adopt the following characterization of a problem, which is widely used in the literature (e.g. Kantowski 1980): A task is said to be a problem if its solution requires that an individual combines previously known data in a way that is new (to him). If he can immediately recognize the measures that are needed to complete the task, it is a routine task (or a standard task or an exercise) for him. If one is not willing to distinguish between a problem and a routine task, one can use the word "task". In this context, problem solving can be understood as "a process where previously acquired data are used in a new and unknown situation" (NCSM 1989). In the following, we use an example to further illustrate our view of the concept of problem. The following task is modified from a problem of the book of Müller & Wittmann (1978, 43). Example 1 (Construct a wall). Add always two neighbouring numbers and put the sum above them (Figure 1). What might be the top number? Example 1 is a routine task for everybody who can calculate with whole numbers. But if we ask the solver to work backwards, i.e. giving only some numbers from between, this might turn out to be a problem


It is worthwhile noting that the concept of problem is relative to the individual and to time. A task that is a problem for one person, might be a routine task for another. A problem of today might become a routine task later. Often in the beginning of problem solving, pupils deal with problems where they need only to have one insight in order to find the solution. Usually the key point is to perceive the problem situation in a new way. These problems are called one-step problems or mathematically simple problems; the wording mathematical puzzle is also used. For example usually matchstick problems are such. In the 1970’s the term ‘investigation’ was introduced in England to mean an extended problem situation. In investigations the starting situation is usually given and, perhaps, the first problems, too, in order to show some possible ways ahead. Pupils are expected to choose their problems and route. For example, the first example above might give a rise for an investigation, as follows: ”What problems can you develop from the situation of Figure 1?” One may separate structured and unstructured investigations. The latter one is the English version. In the continent, the concept of structured investigations was developed (e.g. Pehkonen 1995). He called them problem fields, and characteristic for them was that a teacher has a collection of different connected problems, and he/she may select the way of continuation. For example, in Figure 1 the teacher might ask ”What will happen to the top number (-3), if you change some of the starting numbers (2, 5, -7, 1)? Can you predict?” or ”Can you construct a wall, where the top number is 100 (or -10,5 or �5)?” In the following, the use of problem solving in Finnish mathematics instruction is described. Special attention is given to the way the textbooks deal with problem solving, how problem solving is implemented in school, and in what way problem solving and pupils’ problem solving skills are assessed.

Problem solving in the Finnish curriculum

Curriculum development in Finland has reflected the international trends – usually with a delay

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Figure 1. Construct a wall.


of about 10 years. After the “new math” movement in the late 1960’s and early 1970’s, there was a shift ‘back to basics’ towards the end of the decade. Since the 1980’s a lot of emphasis has been given for problem solving. (Kupari 1999). For more than twenty years, problem solving has been one of the general overall goals in the Finnish curricula (NBE 1985, 1994, 2004). Its implementation has been in the focus of teacher pre-service and in-service education since the end of the 1980’s. In 1986 the National Board of Education made systematic efforts to promote problem solving in school mathematics. It organized a two-part seminar 1986 and 1987 in problem solving for teacher educators. There were lectures and demonstrations, also in a school class, on the use of different problems. The participants were urged to apply these problems in their own teaching and to reflect upon them in the second part of the seminar a year later. In the seminar, the participants' conceptions of problem solving were charted with a questionnaire, and reported later on (Pehkonen 1993). The main results of the study were, as follows: The teacher educators saw problem solving as important, since it helps the fostering of the pupils’ cognitive skills, and helps pupils to use the mathematics they have learned. Giving appropriate tasks was considered as the most important approach to teaching problem solving. Teaching problem solving should be carried out in a creative, flexible and approving manner, and open discussions were perceived a necessity. The teacher should involve pupils in problem solving through letting them solve their own problems and through dealing with problems from their familiar surroundings. Pupils’ willingness to study problem solving was considered the most important prerequisite for teaching problem solving. (Pehkonen 1993) For example, the national curriculum for the comprehensive school (NBE 1994) provided rather general guidelines, and local schools were supposed to plan their more detailed curriculum documents within this framework (cf. Pehkonen & al. 2007). The importance of problem solving is clearly acknowledged in the curricular documents (NBE 1985, 1994, 2004).

Problems in textbooks Before the problem-solving seminars in 1986–87, problem tasks were rather rare in Finnish mathematics textbooks. After the seminar almost every printing house published a set of problems, either as a booklet or as a deck of cards, and with time some problems were taken into the textbooks. But a study (Kari 1991) shows that in the Finnish textbooks for grade 7, the proportion of problem tasks was about 11 % of all tasks. Further non-systematic investigations of Finnish mathematics textbooks by teacher students show that the number of problem tasks has not increased in the last decade. The 1990’s was a very fruitful decade in Finnish mathematics education. The National Board of


Education published a guide book (Seppälä 1994) to help teachers when implementing the curricular framework (NBE 1994). Furthermore, new textbooks (usually three or four competing series) were elaborated and published according to the curricular framework. This time both in the elementary level and in upper level of the comprehensive school, there was a book series that was devoted to train pupils’ problem solving skills. For example, the mathematics book series “Mieti ja laske” [Think and Calculate] (Vähäpassi & al. 1997) provides grade 3 pupils (and their teachers) with one page of exercises, one page of word problems, one page of problems and one page of investigations to choose from for each topic. A mathematics book for grades 7–9 of the comprehensive school “Matka matematiikkaan” [A Journey to Mathematics] (Espo & Rossi 1996) was launched. The focus of this textbook was teaching mathematics via problem solving, i.e. almost all contents were introduced via proper problem situations. On one side the use of the book demanded much preparatory work from teachers’ side, but on the other hand it made mathematics teaching more interesting and for pupils an adventure. But the time seems not to be ripe for such radical textbooks, since teachers were not willing accept them but to stick in traditional mathematics books. It seems that the elementary school book (Vähäpassi & al. 1997) is used without thinking or solving problems, only calculating. And the upper school text book (Espo & Rossi 1996) was selected only by a few teachers, less than 10 %. But the influence of these books can be seen in the next generation of the mathematics books from other printing houses.

Use of problems in mathematics lessons

In the 1980’s, there was much teacher in-service training for teachers of comprehensive school on activating teaching methods and problem solving. These components could be seen also in teachers’ beliefs. Both elementary teachers and mathematics teachers regard problem solving as an important aspect of mathematics teaching (Kupari, 1999). However, results after twenty years show that only some of the teachers have changed their teaching style (ibid). Even teachers who express beliefs favorable to problem solving, often fail to implement it in their own teaching (Perkkilä, 2002). This phenomenon of unsuccessful teacher change has been dealt with in a recently published paper (Pehkonen 2006). A recent study on primary teachers’ conceptions on problem solving (Sivunen & Pehkonen 2008) shows that teachers’ conceptions on problem solving have not much changed within twenty years (cf. Pehkonen 1993). Although the development in mathematics teaching has not been as rapid as expected, there are some changes to be observed. The use of problem solving tasks is quite popular today in mathematics lessons, but mainly in the form of mathematical puzzles. If we use the language introduced by Schroeder & Lester (1989), we might say that only few teachers are teaching via problem solving, while most of them teach something about problem solving. The latter means


that they might use some mathematical puzzles in their teaching or have a problem box in their class or something similar. And the former states that these teachers use problem solving as a teaching method, and that is very rare. AN OPPORTUNITY FOR CHANGE: USING OPEN-ENDED PROBLEMS In the attempts to find a new teaching method that might meet the challenges set by constructivism, the so-called open approach was developed in the 1970’s in Japan (e.g. Becker & Shimada 1997, Nohda 2000). Internationally it is accepted that open-ended problems form a useful tool in the development of mathematics teaching in schools, in a way that emphasizes understanding and creativity (e.g. Nohda 1991, Silver 1993, Stacey 1995). Discussion papers from a sizable group of international specialists have been collected and published in a special report (Pehkonen 1997). What are open problems? Tasks are said to be open, if their starting or goal situation is not exactly given (cf. Pehkonen 1995). In open tasks, pupils are given freedom, possibly even in the posing of the question, but at least in the solving of the task. In practice this means that they may end up with different, but equally correct solutions, depending on the additional choices made and the emphases placed during their solution processes. Therefore, open tasks usually have several correct answers. When open tasks are used in mathematics teaching, pupils have an opportunity to act like creative mathematicians (cf. Brown 1997). In Finland, these ideas have been spread out in teacher in-service courses, in teachers’ journals, and in teacher pre-service education for more than twenty years. The leading idea has been to increase openness and creativity in mathematics teaching. Open problems encompass several types of problems (cf. Pehkonen 1995): investigations (a starting point is given), problem posing (or problem finding or problem formulating), real-life situations (which have their roots in everyday life), projects (larger study entities, requiring independent work), problem fields (or problem sequences or problem domains; collections of contextually connected problems), problems without a question, and problem variations (the ”what-if”-method). Several examples of different types of open problems can be found e.g. in the papers by Nohda (1991), Silver (1995), Stacey (1995), Schupp (2002) and Pehkonen (1997, 2004). Some problem fields as an example As an example of open-ended problems, we will consider a couple of problem fields. These represent a type of problems that the author has developed for Finnish heterogeneous classes of comprehensive schools. In each problem field, the difficulty of the problems ranges from very


simple ones that can be solved by the whole class, to harder problems that only the more advanced students might be able to solve. One key characteristic of problem fields is that they are not bound to a fixed grade or age, but are suitable for mathematics teaching from the primary level to teacher in-service education. The particular role of the easier problems in problem fields is to reinforce the problem solving persistence of the pupils. A very important technical aspect of these problems is the way in which they are introduced to a class: The problem field ought to be given gradually to the pupils, and the continuation should be related to the pupils’ solutions. The mental processes involved in the problem solving are of paramount importance, since the role of the answers and results is played down. One aspect of this is using the pupils' own creative power. Thus, in which direction and to what scope the teacher expands a problem field depends on pupils’ answers. We will consider a couple of examples. Example 2. Number triangle. There is a triangle where the corners are free and certain numbers are fixed on the sides (Fig. 2). What numbers should be placed in the blank circles of the triangle, in order the sum of the three numbers on each side is the same? (Pehkonen 1988)

Figure 2. Number Triangle problem.

Since there are many solutions for this problem, we can continue e.g. with following questions: - Can you find another solution? - How many different solutions could there be? - Is it possible to use negative numbers in the circles? - Can you find a solution where the triangle’s side sum (i.e. the sum of the numbers on the same

side) will be 80? Example 3. Polygons with matchsticks Twelve matchsticks (or cocktail-sticks, etc.) will be needed to concretize the problems. The starting situation is the following:

With twelve matchsticks one can make a square (Fig. 3) the area of which is 9 au (au = area units).

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Figure 3. A square with 12 matches. From this situation a sequence of problems (a problem field) is developed. Firstly, we will choose another area, but have the perimeter of the polygon constant. Thus, in each problem the perimeter of the polygon should be made up of 12 matches. - Can you use twelve matches to make a polygon with an area of 5 au? If we are willing to give more thinking time to slower pupils, the faster ones can be asked to find another (perhaps also a third) solution. Usually pupils will find some solutions, and therefore, we might ponder as the next question the number of different solutions. - How many different polygons of 5 au can you make with twelve matches? Can there be more than ten different solutions? The pupils will probably find many of the solutions. But there are still some complicated solutions that they might not find. The following step might be the comparison of the different solutions found by pupils. How many of all the different solutions can be found when the whole class is working together? Another easier direction to vary the problem is to change the area again. - Is it possible to use twelve matches to make a 6 au (or 7 au, 8 au) polygon? Some solutions can be found easily. But are there any other solutions in each case? And how many different ones? The method of cutting out a corner from a rectangular polygon, as earlier, is successful down to area 5 au. But the question of smaller areas is more complicated, since we are compelled to change the method. - Is it possible to use twelve matches to make a 1 au (or 2 au, 3 au, 4 au) polygon? With the aid of the Pythagorean theorem, one can construct polygons with areas of 4 au and 3 au. It should also be possible to find a general solution: the parallelogram. But the question of different solutions and their number in the case of area 2 au (or 1 au) is according to our experience a really hard, but possible one. Still one extension of the problem field is to ask for polygons with areas greater than 9 au: - Using twelve matches is it possible to make a polygon whose area is greater than 9 au? This seems to be a tough one, since in teacher pre-service and in-service courses the problem has so far not been solved. Teaching experiences. During the last fifteen years, the author has worked through the problem field “Polygons with matchsticks” with many groups of teachers at pre-service and in-service


courses, as well as with some school classes in different countries. The main reasons for introducing problem fields in teacher education have been to describe to teacher students how a problem field can be used and to give them an idea how pupils feel when they work with them. Usually, the duration of work with the problem field has taken about 30 minutes, until the ideas of the group have been used up. In the teacher groups, many polygons with areas of 3, 4, 5, 6, 7, 8 au have been found. But those polygons with areas smaller than 3 au seem to be very complicated to construct. Only in a couple of groups somebody has produced the parallelogram as a general solution. CONCLUDING NOTES Summarizing the Finnish experiences of problem solving in mathematics education, we could state that teachers in Finland are changing in the direction of a more favorable attitude to problem solving. But the use of problem solving in teaching demands much from the teacher, and, therefore, they find excuses why not to use a problem-solving approach. The younger generation of teachers seem to be more self-confident and open to changes. The positive experiences of the use of problem fields are similar to the ones reported e.g. by Liljedahl (2005). As part of a compulsory mathematics course he presented a group of pre-service elementary teachers a set of mathematical problems to solve. Some of the tasks allowed a form of mathematical discovery that he called a 'chain of discovery'. They facilitated a state of sustained engagement and even helped to change the student teachers’ negative beliefs and attitudes. Problem fields, as described here, also allow such chains of discovery and sustained engagement.

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[Problem solving in the textbook of the grade in the beginning of the 1990’s.] University of Helsinki. Department of mathematics (Master’s thesis).

Kilpatrick, J. 1994. Trends in Mathematics Assessment – Especially in USA. In: Ainedidaktiikan teorian ja käytännön kohtaaminen [Meeting of theory and practice in subject didactics] (eds. H. Silfverberg & K. Seinelä), 25–33. Reports from the Department of Teacher Education in Tampere A18/1994.

Kupari, P. 1999. Laskutaitoharjoittelusta ongelmanratkaisuun. Matematiikan opettajien matematiikkauskomukset opetuksen muovaajina. [From practising computational skills to problem solving. Mathematics teachers’ mathematical beliefs and the construction of their


teaching]. University of Jyväskylä. Institute for Educational Research. Research Reports 7. Liljedahl, P. 2004. Sustained engagement: Preservice teachers' experience with a chain of

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NBE 1994. Framework Curriculum for the Comprehensive School 1994. National Board of Education: Helsinki: Valtion painatuskeskus.

NBE 1999. Peruskoulun päättöarvioinnin kriteerit. Arvosanan hyvä (8) kriteerit yhteisissä oppiaineissa. [Criteria for final assessment in the comrehensive school. Criteria for the mark good (8) in common subjects] Opetushallitus. Helsinki: Yliopistopaino.

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Pehkonen, E. 1993. What are Finnish teacher educators’ conceptions about the teaching of prob-lem solving in mathematics? European Journal for Teacher Education 16 (3), 237–256.

Pehkonen, E. 1995. Introduction: Use of Open-ended Problems. International Reviews on Mathematical Education 27 (2), 55-57.

Pehkonen, E. (ed.) 1997. Use of open-ended problems in mathematics classroom. University of Helsinki. Department of Teacher Education. Research Report 176.

Pehkonen, E. (ed.) 2001. Problem Solving Around the World. University of Turku. Faculty of Education. Report Series C:14.

Pehkonen, E. 2004. State-of-the-Art in Problem Solving: Focus on Open Problems. In: ProMath Jena 2003. Problem Solving in Mathematics Education (eds. H. Rehlich & B. Zimmermann), 93–111. Hildesheim: Verlag Franzbecker.

Pehkonen, E. 2006. What Do We Know about Teacher Change in Mathematics? In: Kunskapens och lärandets villkor. Festskrift tillägnad professor Ole Björkqvist (eds. L. Häggblom, L. Burman & A-S. Röj-Lindberg), 77–87. Åbo Akademi, Pedagogiska fakulteten, Specialutgåva Nr 1/2006. Vasa.

Pehkonen, E., Ahtee, M. & Lavonen, J. (eds) 2007. How Finns Learn Mathematics and Science. Sense Publishers: Rotterdam (in print).


Pehkonen, E., Hannula, M. & Björkqvist, O. 2007. Problem solving as a teaching method in mathematics education. In: How Finns Learn Mathematics and Science (eds. E. Pehkonen, M. Ahtee & J. Lavonen). Sense Publishers: Rotterdam (in print).

Perkkilä. P. 2002. Opettajien matematiikkauskomukset ja matematiikan oppikirjan merkitys alkuopetuksessa. [Teachers’ mathematics beliefs and meaning of mathematics textbooks in the first and the second grade in primary school]. University of Jyväskylä.

Rossi, M. & Pehkonen, E. 1995. Tutkimustehtävät ja niiden arviointi peruskoulun matematiikassa [Investigations and their assessment in mathematics instruction of the comprehensive school]. Helsinki: Opetushallitus.

Schroeder, T.L. & Lester, F.K. 1989. Developing Understanding in Mathematics via Problem Solving. In: NCTM Yearbook 1989 (ed. P.R. Trafton), 31-42. Reston (VA): Council.

Schoenfeld, A.H. 1985. Mathematical problem solving. Orlando (FL): Academic Press. Schupp, H. 2002: Thema mit Variationen – Aufgabenvariation im Mathematikunterricht.

Hildesheim: Franzbecker. Seppälä, R. (ed.) 1994. Matematiikka – taitoa ajatella. Yläaste ja lukio [Mathematics – a skill to

think. Lower and upper secondary school]. Suuntana oppimiskeskus 24. Opetushallitus. Silver, E.A. 1993. On mathematical problem posing. In: Proceedings of the seventeenth PME

conference (eds. I. Hirabayashi, N. Nohda, K. Shigematsu & F.-L. Lin). Vol. I, 66–85. University of Tsukuba, Tsukuba (Japan).

Silver, E. 1995. The Nature and Use of Open Problems in Mathematics Education: Mathematical and Pedagogical Perspectives. International Reviews on Mathematical Education 27 (2), 67-72.

Sivunen, M. & Pehkonen, E. 2008. Finnish elementary teachers’ conceptions on problem solving in mathematics teaching. In: Beliefs and Attitudes in Mathematics Education: New Research Results (eds. J. Maass & W. Schöglmann), 88–100. Rotterdam / Taipei: Sense Publishers.

Stacey, K. 1995. The Challenges of Keeping Open Problem-Solving Open in School Mathematics. International Reviews on Mathematical Education 27 (2), 62–67.

Vähäpassi, A. Hartikainen, S., Vaahtokari, A. & Hänninen, L. 1997. Mieti ja laske [Think and Calculate]. Helsinki: Kirjayhtymä.

A PRACTICAL PROBLEM? A person walks up an escalator every day, taking the steps at a rate of 1 step per second. Doing this, they count 20 steps. One day, when in a rush, they take the steps 2 per second. This time the count comes to 32 steps. How many steps are there on the escalator?


Groups and the Mosaics of the Alhambra

The Alhambra The sun lazily flicked at my eye-lids as the bus came to a stop at a roadside café. It was 8 o’clock and time for a wake-up coffee. I came to life and looked out at the rolling countryside of South-ern Spain. Almost in Granada. Some time later, the bus brought us to our destination under the foot of the Alhambra itself. I had heard so much about that famous city, especially its Islamic origin and its fascinating mosaics. Now I would see for myself how its religious artists tried to capture intimations of the infinite through their replicating patterns.

Besides being an object of beauty, what does a mathematician see in this pattern? Which of the familiar isometric transformations of the plane can be distinguished here? The replication of pat-tern indicates that there must be translational symmetry. Indeed, there must be at least two trans-lations in different directions in order to tessellate the plane – which, after all, is the object of the construction. In the example pattern shown above there is no line of reflection – there are no reflection symme-tries but there are other symmetries. Closer inspection of a primitive cell for the pattern above will reveal that rotations of 120o (order 3) may be performed without distorting the pattern – the pattern has rotational symmetry.

Note: The smallest section which may be translated en bloc and thereby tessellate the plane is known as a primitive cell (or translation region). It is the smallest design object which can tessellate the plane by composition of translations ad infinitum – hence the cosmologi-

A courtyard in the Alhambra

A close-up of one of the patterns of mosaics looks like this


cal and theological connection. A section which may be reflected, glide reflected, or rotated to generate the primitive cell is known as a fundamental region. This is the smallest design object which may tessellate the plane by composition of all the symmetries involved in its pattern.

The Group Structure Mathematically speaking the collection of symmetries described above form a group under com-position of transformations. i.e. the set consisting of the identity translation, two translations (at least) in different directions and a rotation of order 3 satisfies all the group properties under com-position of transformations

- there is an identity (the zero translation); - each operation has an inverse; - the set is closed under composition of these operations; - composition of symmetries obeys the associative law.

Because of the connection with the tessellation of the plane this group is known as a ‘wallpaper’ group. In crystallography it is identified as the ‘p3’ group – ‘p’ for ‘primitive cell’, ‘3’ for ‘rotation order 3’ (This is the notation used by the International Union of Crystallographers since 1952).

How many symmetries? There are other patterns to be found in the Alhambra. Do we know what symmetries are in-volved in each of the different patterns? The answer is that in order to tessellate the plane only these following symmetries are possible – translations, rotations of order 2 (180o), rotations of order 3 (120o), rotations of order 4 (90o), rotations of order 6 (60o), reflections and glide reflec-tions. This last symmetry is unusual in that it involves a combination of a reflection followed by a translation – first a reflection across an axis, then followed by a translation parallel to the line of reflection in a ‘following-footprint’ pattern.

How many groups? The next interesting question: how many different sets of symmetries (and thus distinctive tessel-lating structures) can there be and does each set form a group? This question was first answered in 1891 when it was proved (independently by Fedorov (Russia), Schoenflies (Germany), and Barlow (England)) that there are 17 different sets of sym-metries or distinct wallpaper groups. George Polya rediscovered and popularised the idea in 1924.


Patterns which embody the different group structures Examples of patterns which contain sets of symmetries of the plane:

Back to the Alhambra A long-running challenge has been to identify how many of these distinct wallpaper group pat-terns may be found in the Alhambra? During Maths Week 2008 Professor Jose Maria Montesinos from the University of Madrid out-lined how he took on this task and found representatives of each of the 17 wallpaper group pat-terns in the Alhambra. The talk was delivered, appropriately, at the Chester Beatty Library in Dublin which houses many examples of Islamic art.

Theory of Orbifolds The identification of the patterns is also done under the theory of Orbifolds invented by Conway (1992). The codes for this method are given , as well as the classical system of identification, on the chart opposite. In orbifold notation • Numbers indicate the order of rotation involved, • * (asterisk) refers to reflections, • x (miracle) refers to glide translations, and • o (wonder) only contains translations. See: http://jwilson.coe.uga.edu/emt668/EMAT6680.F99/McCallum/WALLPA~1/SEVENT~1.HTM Under the theory of Orbifolds each symmetry is assigned a number or fractional value. The total sum of these features values must be 2 in order that the orbifold Euler characteristic sum is zero. [In three dimensions, V - E + F = 2 for a convex polyhedron (vertices, edges, faces) is known as the Euler Formula. The Euler characteristic for planar graphs is also 2. ]

Translations Reflections Glide reflections Rotations of order 2,3,6 Rotations of order 2,4

These illustrations may be further investigated at http://mathforum.org/geometry/rugs/symmetry/fp.html

http://mathforum.org/geometry/rugs/symmetry/fp.htmlhttp://jwilson.coe.uga.edu/emt668/EMAT6680.F99/McCallum/WALLPA~1/SEVENT~1.HTM


ie Euler characteristic = 2 - features sum = 0 for a wallpaper group. o has value 2 * has value 1 x has value 1 a digit before a * counts as (n-1)/n a digit after a * counts as (n-1)/2n There can only be 17 permutations of the plane symmetries whose feature values sum to 2.

Further investigations and references: http://www.jcrystal.com/steffenweber/JAVA/jwallpaper/J2DSPG.html (Interactive) http://www.scienceu.com/geometry/articles/tiling/symmetry/p4g.html (Animations). http://clowder.net/hop/17walppr/17walppr.html (Animations) du Sautoy, Marcus. 2008; Finding Moonshine; The Fourth Estate, London. This book gives an interesting account of the symmetries found in the Alhambra.

o 2

** 1+1

xx 1+1

*x 1+1

2222 1/2 + 1/2 + 1/2 + 1/2

*2222 1 + 1/4 + 1/4 + 1/4 + 1/4

22x 1/2 +1/2 + 1

22* 1/2 +1/2 + 1

2*22 1/2 + 1 + 1/4 + 1/4

4*2 3/4 + 1 + 1/4

442 3/4 + 3/4 + 1/2

*442 1 + 3/8 + 3/8 + 1/4

3*3 2/3 + 1 + 2/6

333 2/3 + 2/3 + 2/3

*333 1 + 2/6 + 2/6 + 2/6

632 5/6 + 2/3 + 1/2

*632 1 + 5/12 + 2/6 + 1/4

Table showing the 17 permutations of features and their values under orbifold theory

http://www.jcrystal.com/steffenweber/JAVA/jwallpaper/J2DSPG.htmlhttp://www.scienceu.com/geometry/articles/tiling/symmetry/p4g.htmlhttp://clowder.net/hop/17walppr/17walppr.html


The 17 patterns and their identification codes

p1 o pm ** pg xx cm x*

p2 2222 pmm *2222 pgg 22x pmg 22* cmm 2*22

p4 442 p4m *442 p4g 4*2

p3 333 p3m1 *333 p31m 3*3

p6 632 p6m *632

Designs from : http://clowder.net/hop/17walppr/17walppr.html

http://clowder.net/hop/17walppr/17walppr.html


This flow-chart guides through the process of deciding which pattern belongs to which group. [Adapted from http://euler.slu.edu/escher/index.php/Wallpaper_Patterns]

Neil Hallinan, Dublin

Decision chart for identification of a wallpaper group

pmm

cmm

pmg

pgg

p2

cm

Is there a reflection?



Is there a glide reflection?

Are all rotation centres on mirror

lines?

Are all rotation centres on mirror

lines? Are there

reflections in two directions?

Are there mirror lines intersecting at 45o?

p31m

p3m1

p3

p4m

p4g

p6m

p6

Largest rotation order?

YES

NO

YES

YES

YES

YES

YES

NO

NO

NO

NO NO

NO

NO

YES

3 4

6

p4

Is there a glide reflection?


Is there a glide reflection? pm

pg

p1

None

YES

YES

YES

YES

YES

NO

NO

NO

NO 2


http://euler.slu.edu/escher/index.php/Wallpaper_Patterns


Comórtas Sóisearach Matamaitice Éireann 2010 (Irish Junior Mathematics Competition 2010)

This competition is organised by the Irish Mathematics Teachers Association (I.M.T.A.) Eligibility First Year Students 2009/10 Format One set of question papers and answer key will be posted to each participating school some days before the competition date. Each school will be responsible for photocopying the question paper and administering the First Round. First Round Wednesday, March 10th, 2010 Time : 40 minutes Final May 2010 The top students from the First Round may be invited to compete in the Final at venues to be arranged, provided a certain standard is reached. Entrance fee €30 per school (cheques payable to I.M.T.A) If you wish your school to participate please return the completed Registration Form with the fee no later than November 27th 2009 Applications received after this date may not be accepted. Applications should be sent to: Michael D. Moynihan (Mícheál D. Ó Muimhneacháin) Coláiste an Spioraid Naoimh, Bishopstown, Cork. Phone : 087-2860666 / 021- 4870362 (Evenings) email [email protected] Fax : 021 - 4543625 _______________________________________________________________ Registration Form 2010 Name of Teacher:___________________________________________________ School:__________________________________________________________ School address_____________________________________________________ Phone number________________(Home)__________________________(School) School Fax number_____________________ email address___________________ Approximate number of students participating_______________________________


MATHS WEEK 2009

Maths Week Ireland, October 10th to 17th, 2009 is the 4th annual Maths Week. It is an all-island celebration of mathematics. The website http://www.mathsweek.ie/index.html has all the details of public events. Hopefully, your school also participated through raised awareness in the class-room or some more formal event. Dublin: The maths of social networking; iPods; Hamilton Walk; Maths Trails;... Maynooth:Math Circus, ... Dundalk: Turing Machines, ... Cork: Mathemagic, … Galway: Recreational maths from Spain, ... Coleraine: Dr. Maths, … Jordanstown: Maths presentation, … Belfast: Dr. Maths, ... Limerick: Math Morning at NCE-MSTL;... Waterford: Geometer’s Sketchpad, Probability, … There are also many Radio and TV shows with mathematical content surrounding this event.

http://www.mathsweek.ie/index.html


THE IMPORTANCE OF BEING BEAUTIFUL IN MATHEMATICS

Fiacre Ó Cairbre, Department of Mathematics,

NUI, Maynooth

1. Introduction In this article I will discuss beauty in mathematics and I will present a case for why I consider beauty to be arguably the most important feature of mathematics. However, I will first make some general comments about mathematics that are relevant to my discussion. Mathematics essentially comprises an abundance of ideas. Number, triangle and limit are just some examples of the myriad ideas in mathematics. I find from experience in teaching mathemat-ics and promoting mathematics among the general public that it's a big surprise for many people when they hear that number is an idea that cannot be sensed with our five physical senses. Num-bers are indispensable in today's society and appear practically everywhere from football scores to phone numbers to the time of day. One of my favourite football scores, which I refer to in some talks, is the ‘celebrated’ result:

Louth 1-9 v 1-7 Cork in 1957. I will return to this football score later. The reason number appears practically everywhere is because a number is actually an idea and not something physical. Many people think that they can physically see the number two when it's written on the blackboard but this is not so. The number two cannot be physically sensed because it's an idea. Mathematical ideas like number can only be really ‘seen’ with the ‘eyes of the mind’ because that is how one ‘sees’ ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a black-board are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent. The number 2 on the blackboard is purely a symbol to represent the idea we call two. Many people claim they


do not see mathematics in the physical world and this is because they are looking with the wrong eyes. These people are not looking with the eyes of their mind. For example if you look at a car with your physical eyes you do not really see mathematics, but if you look with the eyes of your mind you may see an abundance of mathematical ideas that are crucial for the design and opera-tion of the car. So what is this idea we call two? If one looks at the history of number one sees that the powerful idea of number did not come about overnight. As with most potent mathematical ideas, its crea-tion involved much imagination and creativity and it took a long time for the idea to evolve into something close to its current state around 2500 BC. Here is one way to think of what the number two is:

Think of all pairs of objects that exist; they all have something in common and this common thing is the idea we call two.

One can think of any positive whole number in a similar way. Note that this idea of two is differ-ent from two sheep, two cars etc. The seemingly simple statement that

20+31=51 is actually an abstract statement, since it deals with ideas rather than concrete objects, and solves infinitely many problems (since you can pick any object you want to count) in one go. This illus-trates the incredible practical power of abstraction and many people do not realise that they use abstraction all the time, e.g. when adding. Note that it's not physically possible to solve infinitely many different problems and yet, Hey Presto! it can be done in the abstract in one go. It borders on magic that it can be done. Abstraction essentially means that we work with ideas and also try to deal with many seemingly different problems/situations in one go, in the abstract, by discarding superfluous information and retaining the important common features, which will be ideas. Many people tend to think of ab-straction as the antithesis of practicality but as the above example of addition shows, abstraction can be the most powerful way to solve practical problems because it essentially means you try to solve many seemingly different problems in one go, in the abstract, as opposed to solving all the different problems separately. The latter approach of solving the different problems separately is what people did as relatively recently as less than five thousand years ago by using different physical tokens for counting different objects. For example, they used circular tokens for count-ing sheep and cylindrical tokens for counting jars of oil etc. Nowadays, of course, thanks to ab-straction, we just do it in one go as 20+31=51 and it doesn't matter whether we are counting sheep or jars of oil. Clearly, there are much more advanced examples of abstraction but the 20+31=51 example captures the essential feature of abstraction. See [1] for more on abstraction. These surprises (that number is an idea and addition is an example of abstraction) can actually be


very positive experiences for some people and these surprises don't confuse them; in fact it can change their perception of mathematics for the better and make them more comfortable with other more complicated ideas because they are now already comfortable with one abstract mathe-matical idea, i.e. number. These surprises also enhance the understanding, awareness and appre-ciation of mathematics for many people. Some people also find it fascinating to know that the idea of number was not always known to humans and was actually created by somebody around 2500 BC. As I said above, before 2500 BC the idea of number had not been created and people used different physical tokens to count different objects. Now, back to that pleasing football score:

Louth 1-9 v 1-7 Cork in 1957 Sometimes I use this result, and other examples, to illustrate how number is an idea and why it is so prevalent in today's society. I comment on how the same symbol 9 is used in two different places to indicate two different things. One refers to 9 very satisfying points scored by Louth, while the other refers to 9 hundreds of years. The reason for this is that 9 is just a symbol to rep-resent an idea and that idea can slot into infinitely many different situations. This is one reason why mathematical ideas and abstraction are so powerful and ubiquitous in society today. Some other important features of the above scoreline are that it was the last time that Louth won the All Ireland senior football title, it shows the smallest county defeating the largest county and I could go on! I will now move on to the main topic of this article.

2. Beauty in mathematics The beauty in mathematics typically lies in the beauty of ideas because, as already discussed, mathematics consists of an abundance of ideas. Our notion of beauty usually relates to our five senses, like a beautiful vision or a beautiful sound etc. The notion of beauty in relation to our five senses clearly plays a very important and fundamental role in our society. However, I believe that ideas (which may be unrelated to our five senses) may also have beauty and this is where you will typically find the beauty in mathematics. Thus, in order to experience beauty in mathematics, you typically need to look, not with your physical eyes, but with the ‘eyes of your mind’ because that is how you ‘see’ ideas. From my experience in the teaching of mathematics and the promotion of mathematics among the general public, I have found that the concept of beauty in mathematics shocks many people. However, after a quick example (like the big sum for a little boy below) or two and a little chat


the very same people have changed their perception of mathematics for the better and agree that beauty is a feature of mathematics. One of the reasons why many people are shocked when I mention beauty in mathematics is because they expect the usual notion of beauty in relation to our five senses but as I said above the beauty in mathematics typically cannot be sensed with our five senses. Around 2,500 years ago the Classical Greeks reckoned there were three ingredients in beauty and these were:

lucidity, simplicity and restraint. Note that �simplicity� above typically means �simplicity in hindsight�, because it may not be easy to come up with the idea initially. On the contrary, it may require much creativity and imagina-tion to come up with the idea initially. These three ingredients above might not necessarily give a complete recipe for beauty for everybody, or maybe a recipe for beauty doesn't even exist. How-ever, it can be interesting to have these ingredients in the back of your mind when you encounter beauty in mathematics. Also, for the Classical Greeks, the three ingredients applied to beauty, not just in mathematics, but in many of their interests like literature, art, sculpture, music, architec-ture etc.

3. Some examples of beauty in mathematics Example 1. Big sum for a little boy Here is a simple example of what I consider to be beauty in mathematics. A German boy, Karl Friedrich Gauss (1777-1855), was in his first arithmetic class in the late 18th century and the teacher had to leave for about 15 minutes. The teacher asked the pupils to add up all the numbers from 1 to 100 assuming that would keep them busy while he was gone. Gauss put up his hand before the teacher left the room. Gauss had the answer and his solution exhibits both beauty and practical power. Gauss observed that:

1+100=101, 2+99=101, 3+98=101,

… … …

50+51=101 and so the sum of all the numbers from 1 to 100 is 50 times 101 which is 5050. Notice how Gauss' solution exploits the symmetry in the problem and flows very smoothly. Compare it to the direct brute force approach of 1+2+3+4.... which is very cumbersome and would take a long time. Both approaches will give the same answer but Gauss' solution is elegant and the other is tedious. Gauss' approach is also much more powerful than the 1+2+3... approach because his idea can be


generalised to solve more complicated problems, but you cannot really do much more with the 1+2+3... approach. This power of the beauty in mathematics happens frequently. For those people who are shocked by the notion of beauty in mathematics, this example from Gauss usually changes their perception of mathematics very quickly for the better and they then agree that beauty can be a feature of mathematics. Example 2. The Seven bridges of Königsberg This is the famous Seven bridges of Königsberg puzzle. Königsberg, which is now called Kalin-ingrad in Russia, was a city in East Prussia during the eighteenth century. The city was on the banks of the River Pregel and the four parts of the city, denoted by A, B, C, and D, were linked by seven bridges. See Figure 1

Figure 1 On Sundays people liked to walk around the city and the following question arose:

Is it possible for one to return to their starting point, anywhere in the city, by cross-ing each bridge exactly once?

It's a bit like the Dublin puzzle which asks: “Can you walk from one side of Dublin to the other without passing a pub?� I suppose you could call it ‘The infinite pubs of Dublin’ puzzle! Anyway, nobody could solve the Königsberg puzzle until the famous Swiss mathematician, Euler (1707-1783), heard about the puzzle and solved it in 1736. Euler proved that it's impossible to return to your starting point by crossing each bridge exactly once. So, how did Euler's proof go? Well, suppose for convenience that your starting point is in A. The same argument will work for B, C and D. Now, Euler observed that a necessary condition


for being able to return to your starting point after crossing each bridge exactly once is that there must be an even number of bridges linked to A. The reason for this is that you must leave A on some bridge and then come back to A on a different bridge, then leave and come back again etc. If you think about it you will see that if there was an odd number of bridges linked to A, then you would have no last bridge to come back on. Now, one can see that there is actually an odd num-ber of bridges linked to A and so you cannot return to your starting point after crossing each bridge exactly once. I think Euler's proof is ingenious and is the epitome of elegance. The Classi-cal Greeks would consider it beautiful too because it certainly has those three ingredients of lu-cidity, simplicity (in hindsight) and restraint. Euler's solution is a famous example of elegance in the history of mathematics. Furthermore, his solution of a seemingly trivial puzzle led to a whole new area in mathematics called network the-ory (or graph theory) which is now indispensable for understanding and designing telecommuni-cation networks, computer circuits, complicated timetables (like our university timetable here in Maynooth) and much more. This is a great example of how an elegant solution of a seemingly innocent puzzle can lead to a major breakthrough in mathematics which in turn can produce very powerful solutions to all sorts of important problems in engineering, science and many other ar-eas. Notice that nowadays one could just throw this puzzle at a computer and the computer would just check all the millions of possible routes and conclude that it's impossible to return to your starting point by crossing each bridge exactly once. However, I don't see any elegance in a computer churning out the word ‘Impossible’. The computer approach provides no insight into why it's im-possible and furthermore doesn't give you any new ideas that could be applied elsewhere. How-ever, Euler's approach provides insight into why it's impossible and his idea, as I said above, led to a whole new area in mathematics that is now indispensable for solving many important prob-lems in engineering, science and many other areas. So, maybe it's just as well there were no com-puters in Euler's time! Also, it's interesting to note that there are no longer seven bridges in Königsberg because the city was bombed heavily during the second world war. Only three of the original bridges are left and two of the others have been rebuilt. Apparently, it is now possible to return to your starting point by crossing each bridge exactly once, unlike back in Euler's time in 1736! Euler was the most prolific mathematician ever, in terms of number of publications, until the Hungarian mathematician, Erdös (1913-1996), passed him out recently. Erdös was so prolific that apparently, on a long train journey once, he ended up chatting with the train conductor, who was not a mathematician, and between the two of them they solved a previously unsolved problem and published it later! Euler was not only prolific in mathematics; he also had thirteen children. Actually, he once said that some of his greatest mathematical ideas came to him while he had a sleeping baby on his lap. Note that one could base an outdoor mathematical activity on the


Königsberg puzzle by finding a place near your school, like an appropriate variety of paths in a park, and ask a similar question as in the Königsberg puzzle above. Example 3. The walk along a mountain path This strange looking puzzle seems to have nothing to do with mathematics and yet it is one of my favourite examples of beauty in mathematical thinking. The puzzle goes as follows:

Deirdre starts walking along a mountain path from her house to Ciara's house at 9 a.m. on Saturday morning. Deirdre stays overnight at Ciara's house and starts walking back along the same mountain path at 9 a.m. on Sunday morning. Is there a point on the mountain path where Deirdre passes at the same time on both days?

Notice that there are no assumptions made about Deirdre's speed on either day. She may walk faster on one day than the other; we don't know and it doesn't matter. The solution to this puzzle involves thinking outside the box in a big way. Here is the solution: Imagine the Saturday walk and the Sunday walk starting simultaneously and you will see that the two walks must intersect at some point X. This point X is a point on the path where Deirdre passes at the same time on both days. That's it! Now, that solution has beauty. Example 4. An extraordinary equation The following equation is widely regarded to be the most beautiful equation in mathematics:

Why? Well, essentially because it embraces the five most important numbers in mathematics and

it does so in quite a lucid and relatively simple way. The five numbers all have very different ori-

gins and yet it's quite extraordinary that one relatively ‘simple’ relationship embraces them all.

Each of the five numbers

has a fascinating history. The most interesting book title related to these numbers is undoubtedly

‘Zero: The biography of a dangerous idea’ by Charles Seife. Read it and you will see why zero

was and still is a dangerous idea. Second in the league of interesting book titles for these numbers

is ‘An imaginary tale: the story of ’ by Paul Nahin.

Equation (*) above can be proved by setting in the equation: . Equation (*) is not only very aesthetically pleasing, but it also is very useful in a practical way. For example, it plays a fundamental role in helping us understand how things change periodically

1 0 (*)ie π + =

0, 1, , ,e iπ

1−

θ π= cos sinie iθ θ θ= +


in time. In fact, the electricity supply industry, which utilises alternating current to provide elec-tricity, uses equation (*) and its consequences every time it designs and operates a power station. So, quite literally, in this case mathematical beauty definitely has practical power! Example 5. Magic This example provides a taste of the magic in mathematics. I will present this example in the form of a trick below. Tricks can often be a good way to stimulate students. They can also pro-vide an intriguing setting for the discussion of mathematics. The trick below has many important applications to science. For example, the trick relates to why students can listen to their favourite music on a CD and why a CD supposedly has no flaws/scratches etc. like the old LPs. The trick also relates to why we can view images from Mars! Here is the trick: Create an audience of students. Ask a volunteer to set up a square with five rows and five col-umns of cards (or anything that has a front and a back side that are different), with a random number of cards face up and face down. Ask the volunteer to turn one of the cards over while you are not looking. The trick is that you will be able to say which card was turned over. However, just before the student turns the card over, you suggest adding in one card to each of the five original rows and one card to each of the five original columns in order to make your problem more difficult. This action is crucial to the trick but you don't let the audience know this. You carefully, yet seemingly carelessly, append a new card to each of the five rows and a new card to each of the five columns such that the number of cards face up in each of the first five new rows is even and the number of cards face up in each of the first five new columns is even. You then look away and let the volunteer turn one card over. You look back at the cards (and wave your magic wand!) and simply silently count the number of cards face up in each of the first five new rows and each of the first five new columns and note where you get an odd answer. This will tell you where the overturned card lies. It will seem like magic. The above trick can also be performed by using zeros and ones on the blackboard instead of cards face up and face down. I have performed this trick many times in my public promotion of mathematics in schools and the general public and the trick definitely makes a big impression on people. I feel the idea behind this trick has a certain beauty to it. Where there is beauty in mathematics, practical power will often follow, and so it's not surprising that this idea also has important practical applications. How does the above trick relate to applications in science? Well, in the trick you are using a basic version of a technique that is fundamental in the powerful practical area of ‘error correction in codes’. This is the technique where information is appended to the code (message) by the trans-mitter, in order that the receiver of the message will be able to detect a possible error, due to


physical interference etc, and hopefully correct the error. The analogue of the error in the above trick is the overturned card and you were able to detect where the error lies essentially by append-ing extra information before the card was turned over. Error correction in codes is crucial in the performance of compact discs. Take a CD from your music collection. The sound is digitally stored on the CD. This digital information can be thought of as a code (message) consisting of zeros and ones, just like the face-up cards and face-down cards in the trick above. Extra information is also appended to the CD as in the trick above to give the total code on the CD. A laser beam in your CD player transmits this total code to a de-coder. The decoder receives the total code and attempts to detect any errors which may have been caused by dirt or a scratch etc. This detection process is an advanced version of the method used in the trick above. When an error is detected it can then be corrected so that the sound emanating from your CD player is correct. This is a far cry from the needle on the turntable! Error correction in codes is also fundamental in analysing information transmitted from space-craft. For example, when a spacecraft takes photos of Mars, the information is digitally stored like in the CD above. This information (and the extra appended information like above) is trans-mitted to earth. Any errors caused along the way, like radio interference etc, can be detected and corrected like above. We can then see the correct images of Mars. Example 6. It's a knock out Recall that I said I could go on, in relation to the Louth v Cork scoreline. Well, this example is related to that scoreline. How many games took place in the 1957 All-Ireland senior football championship before the Louth captain, Dermot O'Brien, lifted the Sam Maguire to the cheers of all the jubilant Louth fans at Croke Park? It's not obvious, is it? Here is a similar, yet seemingly more difficult problem: Pick any knock-out tournament you want; it could be football, tennis etc. Suppose there are 127 teams involved and that each game produces one winner who proceeds to the next round and one loser that cannot return to the tour-nament later on. So, there are no draws, replays or GAA-backdoor-like features. How many games must be played before the champion lifts the trophy? Generalise this to the case where you replace 127 by any positive number n. This looks like quite a complicated problem because you don't know if some teams have byes into later rounds and you don't have any information on the structure of the tournament other than what is mentioned above which doesn't seem like enough information. Nevertheless, there will be a beautiful two-line solution to this problem. This is a good example of how, by looking at the problem in a completely different way, the solution just simply pops out. Another feature of this problem is the following metaphor which I sometimes mention in my promotion/teaching of mathematics:


You might feel like you are banging your head against a brick wall and there is no way through to the other side. However, maybe there is an unlocked door somewhere in the brick wall and you just need to gently push it open and there you are, on the other side.

Instead of concentrating on the start of the tournament and looking forward in time, like most people do, we will go to the end of the tournament to produce the elegant two-line solution:

The ‘champion lifting the trophy’ is equivalent to ‘exactly 126 losers’ which is equivalent to ‘exactly 126 games played’! Consequently, the answer is 126.

Similarly, the general solution to the n-team problem is n-1. It's interesting to note that, in the Junior Certificate Syllabus, one of the general objectives in mathematics education is that the students should appreciate mathematics as a result of being able to acknowledge the beauty. There are many other examples of beauty in mathematics. Some re-quire more advanced material and some don't. Here are just two more examples, of many: Euclid's elegant proof that there are infinitely many primes and the aesthetically pleasing proof by Hippasus that is irrational.

4. Why beauty is arguably the most important feature of mathematics From my experience teaching a course on the history of mathematics I feel that beauty in mathe-matics is arguably the most important feature of mathematics. I will present a case for this opin-ion shortly. Five other important features of mathematics are:

a) Deductive reasoning. See Reason (iii) below for more on this.

b) Abstraction. See section 1 above for more on this.

c) The practical power of mathematics, i.e. the powerful applications of mathematics to sci-ence, engineering, navigation, meteorology, finance and many other areas.

d) Research. Historically, research in mathematics has been very vibrant with mathemati-

cians trying to solve many unsolved problems and also developing new theories. The mo-tivation for mathematical research can come from a problem in the physical world or just from pure human imagination. One can play ‘Who wants to be a Millionaire?’ in mathe-matical research! How? Well, go to www.claymath.org and check out the Clay Mathemat-ics Institute's Millennium Problems. There is a million dollars prize money for solving

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www.claymath.org


any of the seven Millennium Problems. Let me know if you solve any and I would be happy to be your agent! One of the unsolved Millennium problems relates to the Navier-Stokes equation, which is partially named after an Irish mathematician. George Stokes (1819-1903) was born in Skreen, Co. Sligo. The Navier-Stokes equation is important in many practical problems including the stability of ships and can also be used to model ocean currents. Coincidentally Stokes did a lot of work on fluid dynamics, related to waves and ocean currents, and now Skreen is close to some of the best waves for surfing in Europe (e.g. Easkey).

e) Freedom. The notion of freedom in mathematics shocks many people. However, as Cantor (1845-1918) once said,

“The essence of mathematics lies in its freedom”. The reason freedom is an important feature of mathematics is because one is free to con-

ceive of any ideas one wants in mathematics. Whether or not these ideas will lead to any-thing interesting or useful is another matter. Historically, the major breakthroughs in mathematics have typically happened because the great mathematicians were free to con-ceive of any ideas they wanted even if they broke with conventions and seemed bizarre to other mathematicians and the general public. Three examples, of many, are the discovery that was irrational by Hippasus in Ancient Greece, the discovery of Non-Euclidean Geometry in the 19th century which liberated geometry and the creation of Quaternions by Hamilton on the banks of the Royal Canal in Dublin in 1843 which liberated algebra from arithmetic. See section 6 for more on Quaternions.

Mathematics is so much more than mere numbers, techniques and formulas. Techniques on their own are usually devoid of stimulation and beauty. The art of doing mathematics may involve any of the following: creativity, imagination, inspiration, ingenuity, surprise, mystery, beauty, intui-tion, insight, subtlety, fun, a wild thought, wonder, symmetry, harmony, aesthetic pleasure, origi-nality, a great sense of achievement, a profound idea, a simple and yet powerful idea, deep con-centration and hard work. As I will outline below, features (a), (b), (c), and (d) above are all intimately related to beauty in mathematics. I will now present a case for why I believe that beauty is, arguably, the most important feature of mathematics. I will give four reasons. Reason (i) The quest for beauty has often been the motivation for why the great mathematicians do research in mathematics. Intellectual curiosity, the quest for beauty and the need to understand and solve important practi-

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cal problems (in science and many other areas) are some of the motivating elements for doing mathematics. From my experience in teaching a course on the history of mathematics, I feel that the search for beauty has often been the motivation for why the great mathematicians do research in mathematics. I will let some of these mathematicians speak for themselves: Ireland's greatest mathematician, William Rowan Hamilton (1805-1865), was also a poet and re-garded “Mathematics as an aesthetic creation, akin to poetry, with its own mysteries and mo-ments of profound revelation”. He also wrote: “For mathematics, as well as poetry, has its own enthusiasm and holds its own communion, with the sublimity and beauty of the universe”. Hardy (1877-1947), once wrote:

“The mathematician's patterns, like the painter's or poet's, must be beautiful, the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics. It may be hard to define mathematical beauty, but that is just as true of beauty of any kind - we may not quite know what we mean by a beauti-ful poem, but that does not prevent us from recognising one when we read it”.

The great French mathematician, Poincare (1854-1912), said:

“The mathematician does not do mathematics because it's useful, he studies it because he delights in it and he delights in it because it's beautiful”.

Somebody once wrote: “Many mathematicians do research out of a desire for mathematical ele-gance and the thrill of exploring the unknown”. Archimedes (287-212 BC) is widely regarded as one of the three greatest mathematicians of all time. The historian, Plutarch, once wrote about Archimedes:

“He, i.e. Archimedes, regarded the business of engineering, and indeed of every art which ministers to the material needs of life, as an ignoble and sor-did activity, and he concentrated his ambition exclusively upon those specula-tions whose beauty and subtlety are untainted by the claims of necessity. These studies, he believed, are incomparably superior to any others, since here the grandeur and beauty of the subject matter vie for our admiration with the co-gency and precision of the methods of proof”.

This is quite a remarkable statement when one considers that Archimedes' mathematics was, and still is, exceptionally powerful when applied to the areas of engineering, science and many other areas. Archimedes and Hamilton are great examples of people who pursued mathematics for its aes-


thetic qualities and yet their mathematics has turned out to be incredibly powerful when applied to science, engineering and many other important practical areas. There are many other examples of such people. The moral of the story here is that the practical power of mathematics can be an offspring of the search for beauty. This leads us on to reason (ii) below. Reason (ii) The practical power of mathematics is often an offspring of the search for beauty in mathematics. See examples 1, 2, 4 and 5 in section 3 where one can see the practical power of the beauty in mathematics. The quest for beauty in mathematics is what has motivated many of the great mathematicians and yet their mathematics has turned out to be incredibly powerful in science and many other areas. Very often this search for beauty in mathematics has led to new ideas and dis-coveries of new theories that have fundamentally changed our understanding of the physical world and are now indispensable in the physical world. It's clear from the history of mathematics that the practical power of mathematics is often an offspring of the quest for beauty in mathemat-ics. For example, in the sixteenth century the Polish mathematician, Copernicus, was convinced that the universe was a systematic harmonious structure framed on the basis of mathematical principles, designed by God. This pursuit for an aesthetic harmonious mathematical structure led Copernicus to his famous heliocentric theory which stated that the earth and the planets revolved around the sun as opposed to the earlier belief that the earth was the centre of the universe with the sun revolving around the earth. Copernicus had no experimental evidence for his theory. The motivation for his theory was purely aesthetic because the mathematics describing the sun-centred universe was more aesthetically pleasing than the mathematics describing the earth-centred universe. Galileo and Kepler would later pursue Copernicus' ideas and provide experi-mental evidence that the earth revolved around the sun. This shocked the world and revolution-ised science and society. As we know, Hamilton's motivation for doing research in mathematics was the search for beauty and yet his mathematics has turned out to be incredibly powerful when applied to science and many other areas. For example, his fundamental theory of dynamics was indispensable for the creation of Quantum Mechanics which is how we now understand the physical world at the mi-croscopic level. Also, his famous Hamiltonian function is fundamental to many aspects of phys-ics. Here is what Hamilton wrote about his new ‘General method of dynamics’ in 1834:

“The difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another; and even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from the reduction of the most complex and, probably, of all researches respecting the forces and motions of body, to the study of one characteristic function, the unfolding of one central relation...”

It's clear that Hamilton didn't care if his new theory had practical applications. The important


point for him is that it had ‘intellectual pleasure’, i.e. aesthetic pleasure. However, his new theory did turn out to have many powerful practical applications later on, e.g. in Quantum Mechanics as mentioned above. Again, here we have practical power being an offspring of the search for beauty in mathematics. In section 6 I also show some of the many powerful applications of Ham-ilton's Quaternions. The Classical Greeks did mathematics for aesthetic pleasure, as we will see below in reason (iii). However, their mathematics has turned put to be exceptionally powerful in the practical world. Two examples, of many, are the ellipse and the parabola. They studied the abstract ellipse inten-sively for aesthetic pleasure and their results were exactly what Kepler needed two thousand years later to show that the orbits of the planets were ellipses with the sun at one of the foci. They also investigated the abstract parabola for aesthetic pleasure and their results later helped Galileo show that projectiles from the surface of the earth followed a parabolic trajectory. This solved a very important practical problem in the seventeenth century, around two thousand years after the Classical Greeks. It's important to realise that in the applications of mathematics to the physical world, and else-where, mathematics does a lot more than solve problems: it can analyse, predict and prescribe: it can provide deeper insight and it can generate and explore new ideas. Mathematics has a rich his-tory and it has played a very significant role in our civilisation. Reason (iii) Mathematics, as we know it today, was essentially born out of a pursuit for aesthetic pleasure and beauty by the Classical Greeks around 600 BC. The two main pillars of mathematics are Deductive Reasoning and Abstraction. Around 600 BC the Classical Greeks essentially created mathematics, as we know it today, based on these two pillars. Also, these two pillars, Deductive Reasoning and Abstraction, appealed to the Greeks for aesthetic reasons, as we will see below. Deductive reasoning works as follows: We start with premises (which are accepted facts) and then we make conclusions with certainty. It's this word certainty that makes deductive reasoning very special and distinguishes it from all other forms of reasoning. Deductive reasoning lies at the heart of a mathematical proof and means that a proof, once done correctly, is eternal. This made deductive reasoning very appealing to the Classical Greeks for aesthetic reasons because they were deep philosophers and found the quest for eternal truths to be aesthetically pleasing. Deductive reasoning is also one of the reasons why mathematics underpins so much of science. Notice that in some cases we don't care whether the premises mentioned above are true or not (e.g. in the ‘proof by contradiction’ approach we want to prove our premises are actually false). Note that ‘potentially uncertain’ things like intui-tion, conjectures, etc. also play an important role in the art of deductive reasoning because they


can give you powerful insight, targets for what to prove etc. Much creativity and imagination can also be involved in the art of deductive reasoning because there is no guaranteed approach that will always work. An example of an old proof that is eternal is Euclid's proof that there are infi-nitely many primes. Euclid's proof is as valid today as when it was first done around 2,300 years ago. Deductive reasoning has been called a celebration of the power of pure reason. See [1] for more on deductive reasoning. See section 1 for a discussion about abstraction. The Classical Greeks were greatly attracted to abstract concepts and ideas which they considered to be eternal, perfect and aesthetically pleas-ing. Concrete physical things were, in their opinion, ephemeral and imperfect. They studied the abstract circle, i.e. the mathematical circle rather than a particular physical circle. However, many of their results about abstract concepts turned out to be fundamental in solving practical problems as we saw above in reason (ii). Note that the abstract circle has no thickness, colour or molecular structure whereas a physical circle does. Even though the abstract circle may be suggested by the physical circle, the Greeks emphasised that the abstract circle and the physical circle were two totally different creatures. Deductive reasoning and abstraction are always around whenever mathematics appears. It's very revealing when you realise that mathematics, as we know it today, was born out of a pursuit of aesthetic pleasure and beauty by the Classical Greeks around 600 BC. Reason (iv) As Keats wrote: Beauty is truth, truth beauty. I feel that this reason is not necessarily as strong as the previous three reasons but I think it's worth mentioning. Deductive reas

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